I suspect that the question under consideration is whether or not $VP=VNP$; this is the problem directly studied by geometric complexity theorists, as I understand their work.  This project is described in some detail [here][1] (this paper is by Burgisser, Landsberg, Manivel, and Weyman, describing work of Mulmuley and related people)--it is aimed at algebraic geometers; so you will likely be comfortable with it.  The description of the complexity problem under consideration is in section 9.

The algebraic $VP$ vs. $VNP$ conjeture is due to Valiant, in [this paper][2] and his paper "Reducibility by algebraic projections" which I can't find online at the moment, unfortunately; these are references [63] and [64] in the paper I link to above.  Valiant is, as I recall, a very clear writer, so hopefully you will find these papers readable as well.

Essentially, Valiant argues that some algebraic properties of the permanent and related varieties should have complexity-theoretic implications; a reasonable heuristic for this might be the many combinatorial interpretations of the permanent.  Unfortunately, as far as I know there are few implications between these algebraic versions of P vs. NP and the problem itself; there are some results assuming GRH.  See e.g. [this paper by Burgisser][3].

Hopefully, this is the algebraic analogue of P vs. NP you were looking for.


  [1]: http://www.math.tamu.edu/~jml/BLMWfinal.pdf
  [2]: http://portal.acm.org/citation.cfm?id=804419
  [3]: http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V1G-43G44RV-6&_user=145269&_coverDate=03/17/2000&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_acct=C000012078&_version=1&_urlVersion=0&_userid=145269&md5=414421f201d7d87fa332ac8f3e6ed759&searchtype=a